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Robust Optimization for Environmental and Energy Planning F. Babonneau, J.-P. Vial, and R. Apparigliato

Abstract Uncertainty is often present in environmental and energy economics. Traditional approaches to optimization under uncertainty, e.g., stochastic programming, chance-constrained programming or stochastic dynamic programming, encounter the most severe numerical difficulties because models in this area are large and complex, already in their deterministic formulation. The goal of the present chapter is to introduce a relatively new field, known as robust optimization, as an alternative to traditional methods and formulations. Through an illustrative example, we suggest ways of putting robust optimization at work in environmental and energy optimization models.

1 Robust Optimization in short Uncertainty is often present in environmental and energy economics. As models in this area are often large and complex, introducing uncertainty with traditional approaches, e.g., stochastic programming [26, 25, 18, 33], chance-constrained programming [23, 22, 34, 36] or stochastic dynamic programming [13], generally leads to numerical intractable model as soon as a relevant representation of uncertainty is sought. F. Babonneau ORDECSYS, 4 place de l’Etrier, 1224, Chˆene-Bougeries, Suisse, e-mail: fbabonneau@ ordecsys.com J.-P. Vial ORDECSYS, 4 place de l’Etrier, 1224, Chˆene-Bougeries, Suisse, e-mail: jpvial@ordecsys. com R. Apparigliato GdF-Suez, Paris, France, e-mail: [email protected]

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F. Babonneau, J.-P. Vial, and R. Apparigliato

The goal of the present contribution is to introduce a relatively new field, known as robust optimization [37], which is an alternative to traditional methods and formulations. The main feature of this approach is that it is does not resort to the calculus of probability, which makes it immune against the curse of dimensionality and computational intractability. In the meantime, robust optimization turns out to be a safe approximation of chance constrained programming [23]. We shall illustrate our presentation through an example, a simplified version in power supply subject to constraints on admissible concentration of pollutants. This will provide a support to the presentation of the main ideas, to the methodology and to the value of the solutions. Dealing with uncertainty raises formidable theoretical and practical modeling problems. On the other hand, almost all solution methods stumble on intractability issues. In the last decade, a new approach has emerged, which provides new ways to attack the problem. It is named Robust Optimization, and works on a new paradigm. To present it in few words only, we shall contrast it with traditional methods, such as stochastic programming, chance-constrained programming and/or dynamic programming. Roughly speaking, traditional methods posit the prerequisite of a welldefined probability model of the uncertainties involved in the problem; they next expand the mathematical programming model of the deterministic version of the problem to incorporate the uncertainty. This last operation usually goes with an increase of complexity that, most of the time, puts the computation of the solution out of reach of the current optimization methods. This phenomenon is sometimes described as the curse of dimensionality and/or computational intractability of the model1 . In contrast, the primary concern of Robust Optimization is to overcome the complexity issue in adopting a non-probabilistic formulation of the uncertainty. The main underlying idea is to start with a much simplified, if not simplistic, description of uncertainty and look for solutions that remain satisfactory for all realizations of the uncertain parameters that are allowed by the uncertainty model. Solutions having this property are named robust. In Robust Optimization, no probability model is assigned to the uncertainty, which makes it possible to avoid expensive computations of multi-dimensional integrals associated with probabilities and expectations. Computing robust solution becomes a numerically tractable operation. At first sight Robust Optimization can be assimilated to a worst case approach on a selected subset of possible realizations of the uncertainty parameters. This may give mixed feelings to people attached to the probabilistic aspect of the problem of interest. Of course, one may object that Robust Optimization has the definite advantage of avoiding the dramatic computational shortcomings of traditional approaches, in particular in multistage problems. Indeed, representing with a minimum of accuracy probabilities as well as computing expectations with multivariate distributions are formidable handicaps in the framework of optimization problems. But, recent results in Robust Optimization [5, 22, 23, 20] offer a more positive view. In few words, these results consist in a lower bound on the probability that the computed 1

In very loose terms, a model is numerically intractable if no method can guarantee that a solution can be obtained in polynomial time.

Robust Optimization for Environmental and Energy Planning

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solution remains robust when the whole set of possible realizations—those in the uncertainty set and those outside of it— is considered. These strong results are obtained at the cost of rather mild assumptions on the probability model of uncertainty (independence, a bounded range and an average value at the middle of the range). Probabilities, which are discarded in the initial model, reappear unexpectedly and give stronger confidence in Robust Optimization. The first robust formulation for an optimization problem with uncertainty parameters has been proposed by Soyster [37] at the beginning of the seventies. The concept was taken over in the nineties by El-Gahoui and Lebret [28] and by Ben-Tal and Nemirovski [9]. Since then, robust optimization has been intensively studied in the literature. The main theoretical contributions are [28, 10, 8, 4, 12, 15, 14]. Robust optimization is operational and useful for a large set of decision problems with uncertainty. One can mention decision problems in finance and risk measure [17, 19, 21, 30], in supply chain and inventory management [1, 6, 16], in telecommunications [35] and in management of electricity production in hydraulic valley [2, 3]. This approach is also adapted in complement of optimization techniques such as constraints in probability [20], dynamic optimization [32], or stochastic optimization [24]. The chapter is organized as follows. In Section 2, we present our illustrative example and show that its deterministic solution performs poorly in an uncertainty context. Section 3 is concerned with an application of the robust optimization concepts developed in the next sections to the illustrative example with static uncertainty. In Section 4, we introduce those basic concepts of robust optimization in a static framework. In Section 5, we examine the dynamic case and propose the concept of Linear Decision Rules to cope with the adaptive nature of decisions. Section 6 is devoted to applications to the illustrative example, when the demand over time is the source of uncertainty. We compare the robust optimization solution with a stochastic programming approach. In Section 7 one explores a hybrid approach mixing stochastic programming and robust optimization. In Section 8 we show that bounds on the probability that the robust solution satisfies the constraint with uncertain coefficients can be obtained at the cost of a mild assumption on the probability distribution of the uncertain coefficients. This result shows an obvious relationship between chance constrained programming and robust optimization. Finally, in Section 9 we present an extension of the robust optimization concept that covers both the cases when the uncertainty lies in the uncertainty set and when it lies out of it. A short conclusion discusses possible issues in dealing with practical models in environmental and energy planning that are necessarily of much larger size than the illustrative example of this note. It also gives hints on new developments in the field of robust optimization.

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F. Babonneau, J.-P. Vial, and R. Apparigliato

2 An example in power supply under pollution constraint To help the reader on the concepts of robust optimization that will be presented in this chapter, we illustrate them on a simple environmental and energy planning problem, inspired by the MARKA-Geneva model [27]. The energy planning side concerns the simulation of the evolution of the energy system for a region. The model is based on the assumption of an optimal use of the resources and the simulated planning is obtained by minimizing a welfare function. The energy planning model is enriched to account for a cap on pollutant level in the various subregions. The pollutant level in a given subregion results from local emissions by energy producing sources and from transfers from other subregions. The model is dynamic with three periods. In this model, we consider two types of uncertainties. One is related to the pollutant transfer rates from one subregion to another; the other one concerns the demand. Those two sources of uncertainties are of different natures. We model the demand as an auto-regressive process: a high demand at a time period is an indication that the demand will also be relatively high in the next time period. The decision process must make use of this information and be adaptive. The pollution effect does not enjoy this property. The transfer rates are not stable, but they do not follow any evolution pattern over time. The process is described by an i.i.d. process. In the rest of the chapter, we shall use the terminology region for subregion.

2.1 Deterministic formulation The problem we consider here is the planning of energy production under environmental constraints. The model has four regions, three periods, and three technologies TEM. We denote R = {r1 , r2 , r3 , r4 } the set of regions, T = {t1 ,t2 ,t3 } the set of time periods and P = {p1 , p2 , p3 } the set of production technologies. The model assumes a single output with production target 10, 12 and 14 for the periods t1 , t2 and t3 respectively. The company is committed to serve the demand at all times. Nevertheless, delivery failures may occur if the demand is higher than the production capacity, but the shortage costs, presumably very high, are not known. Thus violations on the demands should be avoided by all means. Three technologies can be used to produce the output. The admissible production level for each technology depends on the installed capacity. This installed capacity depends on installations performed prior to the planning horizon, thereafter named residual capacity, and on investments in the successive periods. These technologies can be installed in four different regions. The residual capacities are given in Table 2.

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The total system cost is the sum over the three periods and the four regions of the capacity investment, maintenance and operation costs for the three technologies. The costs are reported in Table 1. Cost

r1 r2 r3 r4

Maintenance (Mr ) 1 1 1 1 Investment (Ir ) 5 3 4 5 Operation (Or ) 2 2 3 2 Table 1 Maintenance, investment and operation costs for the three technologies.

Technology p1 Technology p2 Technology p3

r1 r2 r3 r4

t1 t2

t3

t1 t2

t3

t1 t2

t3

1 -

-

0.5 -

-

1

-

1 -

-

1

Table 2 Residual capacities (res p,t,r ).

The technologies have different pollutant emission rates, given in Table 3, in each region. A pollution transport and dispersion process takes place. A source-receptor matrix specifies, for each source location, the proportion of the emitted pollutant that is deposited in the different receptor locations (see Table 4). These data are used to determine the level of concentration (immission) in each region. A standard environmental quality, Q = 1.5, is imposed on this concentration for all regions and for all periods. r1 r2 r3 r4 p1 0.7 0.8 0.8 0.6 p2 0.5 0.4 0.7 0.7 p3 0.8 0.9 0.6 0.7 Table 3 Emission rates (E p,r ) by production technology and by region.

The model gives conditions under which a joint investment and production plan meets the demand needs and the environmental standards at minimal cost. For the model, we define the following variables: • x p,t,r : production of technology p, at period t and in region r. • y p,t,r : capacity investment of technology p, at period t and in region r.

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r1 r2 r3 r4

F. Babonneau, J.-P. Vial, and R. Apparigliato r1

r2

r3

r4

0.5 0.1 0.09 0.05

0.1 0.4 0.05 0.1

0.1 0.04 0.5 0.1

0.05 0.1 0.1 0.6

Table 4 Source-receptor transfer matrix (Gri ,r j ).

• z p,t,r : installed capacity of technology p, at period t and in region r. • et,r : emission at period t and in region r. The optimization problem is given by min

x≥0,y≥0,z,e

∑(Or ∑ x p,t,r + Ir ∑ y p,t,r + Mr ∑ z p,t,r ) r

p,t

p,t

z p,t,r = res p,t,r + ∑ y p,τ,r

(1a)

p,t

∀t ∈ T , ∀p ∈ P, ∀r ∈ R

(1b)

τ≤t

x p,t,r ≤ z p,t,r

∀t ∈ T , ∀p ∈ P, ∀r ∈ R

(1c)

∑ x p,t,r = dt

∀t ∈ T

(1d)

p,r

∑ E p,r x p,t,r = et,r

∀t ∈ T , ∀r ∈ R

(1e)

∑ et,ρ Gρ,r ≤ Q

∀t ∈ T , ∀r ∈ R.

(1f)

p

ρ∈R

In that formulation, the equality constraints (1b) express the total installed capacities, z p,t,r , as the sum of the residual capacities and the capacities that are installed in the previous periods. Inequality constraints (1c) limit production to the installed capacities. Equations (1d) are demand constraints. The last two sets of constraints firstly compute the pollutant emissions resulting from all technologies (1e), and secondly impose bounds on the final emissions resulting from the dispersion process (1f). Problem (1) has 108 variables and 99 constraints. Note that (1) can be formulated in a more compact way by removing the z and e variables and the equality constraints (1b) and (1e) (see Subsection 6 for more details). Solving problem (1) gives the optimal objective value 162.056. The optimal investments, productions and emissions are reported in Table 5, 6 and 7, respectively. We observe in Table 5 and Table 6 that the optimal solution is to invest in technology p2 only, except in period t3 where technology p3 is also selected. As far as production is concerned, the three technologies are used in the first two periods, but the first production technology is abandoned in the last period. Table 7 shows that only two emission constraints are active at the optimum (in region r2 at periods t2 and t3 ), and one is close to be active (in region r3 at period t3 ). Should the optimal

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t1

t2

p1 p2 p3

p1 p2 p3

p1 p2

r1 r2 r3 r4

- 1.41 - 6.09 - - - - -

- 0.77 - 1.73 - - - - -

- - 0.15 - - 3.85 - -

Sum

- 7.50 -

- 2.50 -

- 0.15 3.85

7.50

2.50

Total per period

t3 Total

p3

2.18 7.97 3.85 0

4.00

Table 5 Optimal investment schedule. t1 p1

p2

t2 p3

p1

p2

t3 p3

p1

p2

p3

r1 r2 r3 r4

1.00 1.91 - 6.09 - 1.00

1.00 2.18 - 7.82 - 1.00

- 2.18 - 7.97 - 3.85 -

Sum

1.00 8.00 1.00

1.00 10.00 1.00

- 10.15 3.85

10.00

12.00

Total per period

Total 8.27 21.88 3.85 2.00

14.00

Table 6 Optimal production schedule.

r1 r2 r3 r4

t1

t2

t3

1.16 1.21 0.33 0.75

1.24 1.50 0.37 0.82

1.07 1.50 1.39 0.60

Table 7 Emission associated with the optimal solution.

investment/production plan be implemented within an uncertain environment, then, those constraints would potentially be critical and one should expect violations for some variations in the entries of the source-receptor transfer matrix.

2.2 Uncertainties in the power supply model In the power supply model, we consider two sources of uncertainties, one on the pollutant diffusion coefficients and the second one on the demand. First we define the uncertainty on the coefficients of the source-receptor matrix G. Let ξ be a random variable with values in the interval [−1, 1], each coefficient of the matrix G is given by

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F. Babonneau, J.-P. Vial, and R. Apparigliato

Gi, j (ξ ) = G¯ i, j + Gˆ i, j ξi, j ,

(2)

where G¯ i, j is the average coefficient reported in Table 4 and Gˆ i, j corresponds to the coefficient variability. Here we set Gˆ i, j = 0.1G¯ i, j . We now focus on the demand uncertainty. We adopt the following autoregressive model dt+1 = dt + αt+1 + dˆt+1 ηt+1 , where ηt+1 is a random variable with values in the interval [−1, 1] and αt+1 is a deterministic trend factor. This formula can be explicited using backward substitution. Each demand dt appears then as a function of the past random factor ητ for τ = 1, . . . ,t. In our case with horizon of length 3, we have d1 = d¯1 + dˆ1 η1 d2 = d¯2 + dˆ1 η1 + dˆ2 η2

(3b)

d3 = d¯3 + dˆ1 η1 + dˆ2 η2 + dˆ3 η3

(3c)

(3a)

where d¯i and dˆi are the average demand and the variability of the demand, respectively. We have d¯1 = 10, d¯2 = 12, d¯3 = 14 and dˆi = 0.1d¯i , ∀i.

2.3 Validation process The validation process is an empirical study of the behavior of the solution under a set of simulated values for the uncertain parameters. In our study, an item of the simulation process is a set of independent realizations of the random factors (demand and/or emission transfer parameters) over the three periods. We shall use the terminology “scenario” to name one such set of realizations. To generate a scenario, one needs a probabilistic model of the random factors. In our illustrative example, we choose to have the underlying factor to be i.i.d. with a uniform distribution on the range [−1, 1]. The validation is performed on a sample of scenarios of size 1000; each scenario in the sample is a multidimensional vector of realizations of the random factors η and ξ . The demands and the pollutant diffusion coefficients are computed via (3) and (2), respectively. In the evaluation of performance, we must differentiate among the two types of uncertainties. Let us start with the emission transfer coefficients. Since there is no recourse associated with this type of uncertainty, the investment/production can be implemented as such, without a risk of violating the demand constraint. Of course, the constraint on the air quality may not be satisfied, but this is just recorded without any modification of the solution itself. The case of an uncertain demand raises a new issue. One could just record the violations of the demand constraint from above or from below as argued in the pre-

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vious case, but this would not be realistic. Since the production is adapted to the manifested demand, it is difficult to stick to the view that the production should be maintained at its scheduled value if the demand turns out to be smaller than the production. In the validation process we adopt the following strategy. Let x p,t,r and y p,t,r be the production and the investment in a particular solution. If the total planned production ∑ p,r x p,t,r at period t is less than the demand dt , the production is kept as such, and we record a shortage. The ensuing constraint violation is measured in relative value (max{dt − ∑ p,r x p,t,r , 0})/dt . If the demand is less than the planned production, production in each region with each technology is uniformly downsized by the common factor dt / ∑ p,r x p,t,r . The treatment of the constraint on the pollutant level in each region is simpler. It measures the amount of relative violation.

2.4 Evaluation of the deterministic solution in the uncertain environment We now subject the deterministic optimal solution computed in the previous section (see Tables 5, 6 and 7) to the simulation process. Impact of uncertainty on pollutant diffusion Table 8 gives the simulation results with uncertainty on the pollutant diffusion coefficients. We observe that for only 46.5% of the simulations all the air quality constraints are satisfied. For 4.2%, 47.8%, and 1.5% of the cases, one, two and three air quality constraints are violated, respectively. The average and maximum relative violations are about 4.3% and 9.3%, respectively. An analysis of the constraint viAverage relative violation (%) Maximum relative violation (%) % of satisfaction % of one violation % of two violations % of three violations

4.3 9.3 46.5 4.2 47.8 1.5

Table 8 Simulation results with uncertain source-receptor matrix.

olations has revealed that the phenomenon occurred in the three pairs (region and periods) that were critical or near critical in the deterministic study. In the other pairs there is enough slack to absorb variations in the transfer rates. Impact of uncertainty on the demand In Table 9, we report the simulation results conducted on the optimal deterministic solution with a demand uncertainty.

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F. Babonneau, J.-P. Vial, and R. Apparigliato

Predicted cost performance Observed cost performance

162.056 160.871

Scenarios with demand violation(s) in % Conditional average relative violation in % Average number of violations per scenario

62.0 2.5 2.0

Table 9 Simulation results with uncertain demands.

Contrary to the previous experiment, the uncertainty on the demand has an impact on the cost performance. As a result we distinguish between the predicted performance (the optimal cost in the deterministic model) and the observed performance (the average cost in the simulation). Surprisingly enough, the observed performance is better than the predicted one. The paradox is apparent and the explanation is straightforward. When the demand is lower than the average value used in the deterministic model, the production levels are scaled down to match the actual demand and the production costs less than predicted by the deterministic plan. But when the demand is higher than this average, the production cannot be increased, because the capacity constraints in the deterministic optimal solutions are tight; the production costs remain as computed in the deterministic model, but part of the demand is not satisfied. Thus, the seemingly improved cost performance must be balanced with demand violations. In that respect, we observe a demand violation in 62% of the scenarios and a conditional average of unsatisfied demand of 2.5%. Finally, when a violation occurs in a scenario the average number of periods with violation is approximately 2.0 (over the total of 3). The conditional average demand violation is small with respect to the average demand, but one should recall that the range of variation of the demand around the average is ±5%, ±10% and ±15%, in periods 1, 2 and 3, respectively.

3 Case study: robust solution to a problem with uncertain pollutant transfer coefficients To motivate robust optimization, we propose a simple fix to handle uncertainties in the power supply problem. We limit this illustration to uncertainties on the pollutant diffusion coefficients, and consider that the demands are fixed. We shall use heuristic arguments only and postpone theoretical justifications to later sections. Let us start with a general formulation of a linear constraint with uncertain coefficients n

∑ a˜ j x j ≤ b,

j=1

(4)

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where a˜ j are uncertain. For the sake of a simpler presentation, we assume b to be certain. We further describe the uncertain coefficients as linear functions of an underlying random factor ξ a˜ = a¯ + Pξ , where ξ ∈ Rm and P is an n × m matrix. We further assume that the random factor has a symmetric distribution with mean 0. The certain vector a¯ is usually named the normal factor. We can thus focus on the uncertain component of the constraint a¯T x + (PT x)T ξ ≤ b. |{z} | {z } certain uncertain We now evoke a common sense engineering approach that consists in replacing the uncertain term by a safety term κ > 0. By taking a large enough safety term, we give a sufficient guarantee that the solutions of a¯T x + κ ≤ b will almost always remain feasible to (4). This says nothing on the critical way to choose the safety factor, but many practitioners will be receptive to a so-called 2σ , 3σ , possibly 6σ approach. This industrial practice is often justified through the following probabilistic and statistical argument. Assuming that the underlying random factor ξ is a vector with probabilistically independent components with mean 0 and standard deviation σ , we can state that the uncertain component (PT x)T ξ is a random variable with mean 0 and standard deviation ||PT x||2 . In the k × σ approach, the safety term is κ = k||PT x||2 , with the implicit argument that for most distributions a large enough k, say k = 3, will guarantee that a solution to a¯T x + k||PT x||2 ≤ b will satisfy the probabilistic constraint ∑nj=1 a˜ j x j ≤ b with high probability. The two salient features of this “engineer-like” approach can be summarized as follows. An uncertain constraint is replaced by a deterministic constraint with a safety term, and the safety term depends on the decision variables x. Applying this idea to our problem of interest, we obtain that the robust equivalent of the air quality constraints (1f) is

∑ e p,ρ G¯ r,ρ + k|| ∑ e p,ρ Gˆ r,ρ ||2 ≤ Q ρ∈R

∀p ∈ P, r ∈ R.

(5)

ρ∈R

This formulation does not increase the problem size, but the linear air quality constraints are replaced by nonlinear ones. Those constraints are convex conic quadratic, and problems with such constraints can be solved very efficiently by modern solvers (e.g., the open source code [38]). If we wish to remain in the realm of linear programming, we have to replace the safety factor by terms that are amenable to linear inequalities. The following bound will be established in the next section √ ||α||2 ≤ min{ m ||α − β ||∞ + ||β ||1 }, β

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F. Babonneau, J.-P. Vial, and R. Apparigliato

where ||α||1 = ∑m i=1 |αi | and ||α||∞ = maxi |αi |. Hence, the alternative, linear but more restrictive formulation, √ ∑ e p,ρ G¯ r,ρ + k m || ∑ e p,ρ Gˆ r,ρ − β ||∞ + k||β ||1 ≤ Q ∀p ∈ P, r ∈ R. (6) ρ∈R

ρ∈R

In this experiment, we solve the robust equivalent problem with the two formulations, the first one (5) with the conic quadratic constraint, and the other one (6) with linear constraints. We use the same k in the two formulations, but make this k vary to experiment with different degrees of safety. We shall compare the behavior of the robust solutions with the deterministic solution. The results are reported in Table 10.

Cost performance Conditional average of relative violation (%) Maxi relative violation (%) proportion of scenarios with violation (%) proportion of scenarios with one violation (%) proportion of scenarios with two violations (%) proportion of scenarios with three violations (%)

Deterministic

Robust k = 0.8 k = 1 k = 1.2

162.06

163.76 164.21 164.64

4.3 9.3 53.5 4.2 47.8 1.5

1.0 3.0 25.4 17.2 6.4 1.8

0.5 1.5 6 5 1 -

0 -

Table 10 Variable transfer coefficients: behavior on the sample of 1000 scenarios of a robust solution with the conic quadratic formulation (5).

In the second set of experiments we use the formulation with linear constraints (6). We shall prove in the next section that (6) is equivalent to a system of linear inequalities. It turns out that the robust formulation of a typical air quality constraint (1f) corresponding to the pair (t ∈ T , r ∈ R) is

∑ et,i G¯ r,i + k( ∑ ut,r,i + 2vt,r ) ≤ Q i∈R

ut,r,ρ 0 + vt,r ≥

(7a)

i∈R

∑ et,ρ Gˆ ρ 0 ,ρ

∀ρ 0 ∈ R

(7b)

ρ∈R

u ≥ 0, v ≥ 0

(7c)

The above robust counterpart is obtained from the set of inequalities (10) in Proposition 1 (Section 4). The reader will notice that (10) also includes constraints like (7b), but with right-hand side − ∑ρ∈R et,ρ Gˆ ρ 0 ,ρ . In this particular problem, we know that ∑ρ∈R et,ρ Gˆ ρ 0 ,ρ ≥ 0, which makes one half of the constraints redundant. So, we eliminate them and obtain a robust equivalent model that is still linear but with 60 additional variables u’s and v’s and 48 additional constraints.

Robust Optimization for Environmental and Energy Planning

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We solve the robust equivalent problem with different values of k and we report the results of the simulations for each robust optimal solution in Table 11. Deterministic

Robust k = 0.6 k = 0.8 k = 1

162.06

163.49 163.99 164.49

Cost performance Conditional average of relative violation (%) Max relative violation (%) proportion of scenarios with violation (%) proportion of scenarios with one violation (%) proportion of scenarios with two violations (%) proportion of scenarios with three violations

4.3 9.3 53.5 4.2 47.8 1.5

1.2 3.5 29.1 14.9 12.7 1.5

0.7 1.7 12.4 9.4 2.9 0.1

0 -

Table 11 Variable transfer coefficients: behavior on the sample of 1000 scenarios of a robust solution with respect to the linear formulation (7).

t1 p1

t2

p2 p3

p1

p2 p3

t3 p1 p2

p3

r1 r2 r3 r4

- 1.39 - 5.94 0.17 -

-

- 1.51 - 0.91 - 0.08 - -

- 0.10 - 0.16 - - 3.74 - -

Sum

0.17 7.33

-

0.08 2.42 -

- 0.26 3.74

Total per period

7.50

2.50

Total 3.00 7.01 3.74 0.25

4.00

Table 12 Robust investment schedule for k = 0.8 and the linear formulation (7).

The results of Table 10 and Table 11 are very much alike. Nevertheless, we notice that with the same safety level k, say k = 0.8 for a typical example, the robust solution with respect to the linear approximation (6) achieves a slightly worst cost objective (163.99 vs. 163.76) than the solution with respect to the conic quadratic formulation (5). In the meantime, the first solution achieves a better protection against constraint violation. These observations conform to the fact that for identical k the linear safety factor in (6) in Table 11 is an upper bound of the conic quadratic safety factor (5) in Table 10. The first one is more constrained is thus more constrained. The cost performance in not so good, but it ensures a lesser protection against violations. The chances are that in a simulation we observe a lesser number of constraint violations.

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F. Babonneau, J.-P. Vial, and R. Apparigliato

4 Robust Optimization for the static problem: theoretical developments Let us recall the formulation of a linear constraint with uncertain coefficients n

∑ a˜ j x j ≤ b.

j=1

The first basic assumption on the uncertain parameters is that they depend on some random factor ξ in a linear way. Assumption 1 The uncertain vector a˜ is written as a˜ = a¯ + Pξ , where ξ ∈ Rm and P is an n × m matrix. The certain vector a¯ is usually named the normal factor. We can thus focus on the uncertain component of the constraint a¯T x + (PT x)T ξ ≤ b. |{z} | {z } certain uncertain

(8)

Our present goal is to use information on ξ to build the safety factor introduced in the previous section (see inequalities (5) and (6)). The idea is to focus on a subset of all possible events that it is made of all realizations of the underlying uncertain factor ξ that the modeler deems necessary to protect against. This is the so-called uncertainty set. The robust version of the initial uncertain constraint a˜T x ≤ b consists in enforcing the uncertain constraint (8), not for all possible realizations, but only by those in the uncertainty set; that is, the less restrictive constraint a¯T x + (PT x)T ξ ≤ b, for all ξ ∈ Ξ , where Ξ ⊂ Rm is the uncertainty set. A solution to this constraint is called robust with respect to Ξ . If Ξ is a continuous set, the robust constraint is a short-hand writing of an infinite number of simple linear constraints. This seems to put the whole approach into the realm of semi-infinite programming, making the computation of solutions a real issue. It turns out that this is not so, for a large variety of uncertainty sets. We shall consider a few different types of uncertainty sets. Let us start with the ellipsoidal uncertainty set Ξ = {ξ | ||ξ ||2 ≤ k}. In the rest of the chapter, we shall describe the ellipsoidal uncertainty set as the ball in the 2-norm, centered at the origin and with radius k. We denote it B2 (0, k). We

Robust Optimization for Environmental and Energy Planning

15

shall also use more general balls, such as B p (0, k) = {ξ | ||ξ || p ≤ k} p 1/p . The following lemma gives the clue with 1 < p < ∞ and ||ξ || p = (∑m i=1 |ξi | ) to the replacement of the robust constraint by a single finite-dimensional constraint which will be named the robust equivalent.

Lemma 1. Let 1 < p < ∞ and let q be such that

1 p

+ q1 = 1. For any d

max{d T ξ | ||ξ || p ≤ k} = k||d||q . Proof. The proof in the general case is simple but tedious: we omit it. We just illustrate the case of the 2-norm. (See Figure 1.) 

d d/||d||

The maximum is achieved at ξ ∗ = d/||d|| and takes the value dT ξ ∗ = dT

d = ||d||. ||d||

Fig. 1 Maximum of a linear form over a ball in the 2-norm.

4.1 Robust equivalent: the case of the 2-norm We can now state the robust equivalent of the robust constraint. Theorem 1. The robust equivalent of the constraint a¯T x + (PT x)T ξ ≤ b, for all ξ ∈ Ξ = {ξ | ||ξ ||2 ≤ k}, is a¯T x + k||PT x||2 ≤ b. Proof. The proof follows directly from the above lemma. It suffices to replace d by PT x. 

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F. Babonneau, J.-P. Vial, and R. Apparigliato

The factor k plays a crucial role in Theorem 1. The larger its value, the greater the number of realizations ξ against which a solution of a¯T x + k||PT x||2 ≤ b is immunized in the constraint a¯T x + (PT x)T ξ ≤ b with uncertain coefficients . In the sequel we shall use the terminology immunization factor k, or immunization level k. At this stage we should raise some fundamental issues: 1. Is the nonlinear formulation of the robust equivalent a potential source of complexity from a numerical point of view? 2. Should, or could, one consider alternative uncertainty sets? 3. What is the value of starting with the new concept of uncertainty set if one ends up with the same formulation of an engineering safety factor? We shall answer the first two questions, leaving the answer of the last question to the final section of the chapter. Let us start with the first question. It is easy to show that the new constraint is convex (a property of the 2-norm). Moreover, it can be reformulated as a¯T x + kz ≤ b

(9a)

T

P x=u

(9b)

||u||2 ≤ z.

(9c)

The last constraint is conic quadratic, a feature that modern convex optimization codes handle about as efficiently as a linear constraint. Taking the uncertainty set as the primary concept shifts the focus on numerically tractable robust equivalent. However, people concerned with a probabilistic approach may feel ill at ease with the apparent arbitrariness of the uncertainty set. However, there is a powerful theorem in probability theory that assets that under a very mild probabilistic assumption on the random factor ξ , one can provide a surprisingly strong lower bound on the probability that the initial uncertain constraint be satisfied by a robust solution (i.e., a solution to the robust equivalent). The answer to the second question motivates the next section.

4.2 Robust equivalent: the case of the `1 and `∞ norms For some reasons, e.g., a lack of access to a conic quadratic solver, one may want to remain in the realm of linear programming. This can be achieved by resorting to polyhedral uncertainty sets. To link the results with the previous study with ellipsoidal uncertainty sets, we resort to an approximation of the unit ball in the 2-norm by the intersection of balls in the 1-norm and the infinity norm.

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17

The `1 and `∞ norms are natural extensions of the p-norm with p = 1 and p = ∞. It generates the balls m

B1 (0, k) = {ξ | ∑ |ξi | ≤ k} i=1

and B∞ (0, k) = {ξ | max |ξi | ≤ k}. i=1,...,m

Both are polyhedral sets that can be represented by simple inequalities. The next lemma extends Lemma 1 to the two limit cases p = ∞ and p = 1. Lemma 2. We have max{d T ξ | ||ξ ||∞ ≤ k} = k||d||1 and max{d T ξ | ||ξ ||1 ≤ k} = k||d||∞ . √ The B1 (0, k m) and B∞ (0, k) balls can be jointly used to approximate the B2 (0, k) ball in the m-dimensional space as it is illustrated in Figure 2. We can now give the

√ Fig. 2 The set B1 (0, 2) ∩ B∞ (0, 1)

robust equivalent relative to the uncertainty set Ξ = B1 (0, k1 ) ∩ B∞ (0, k∞ ). Lemma 3. We have max{d T ξ | ξ ∈ B1 (0, k1 ) ∩ B∞ (0, k∞ )} = min{k1 ||d − w||∞ + k∞ ||w||1 }. w

ξ

Proof. Let z∗ = max{d T ξ | ξ ∈ B1 (0, k1 ) ∩ B∞ (0, k∞ )}. ξ

Let us replace B∞ (0, k∞ ) by linear inequalities in the above maximization problem. We obtain the alternative expression

18

F. Babonneau, J.-P. Vial, and R. Apparigliato

max d T ξ ξ

−k∞ e ≤ ξ ≤ k∞ e ξ ∈ B1 (0, k1 ), where e is the vector of all ones of appropriate dimension. We form the partial Lagrangian L(ξ , u, v) = d T ξ + uT (ξ + k∞ e) + vT (−ξ + k∞ e) = k∞ eT (u + v) + (d + u − v)T ξ . Clearly z∗ =

max

min L(ξ , u, v)

ξ ∈B1 (0,k1 ) u≥0,v≥0

and by linear programming duality z∗ = min

max

u≥0,v≥0 ξ ∈B1 (0,k1 )

L(ξ , u, v).

The inner minimization problem is L (u, v) = max{k∞ eT (u + v) + (d + u − v)T ξ | ξ ∈ B1 (0, k1 )} ξ

By Lemma 2 we have L (u, v) = k∞ eT (u + v) + k1 ||d + u − v||∞ . One easily checks that an optimal solution of the minimization minu≥0,v≥0 L (u, v) satisfies the complementary condition ui vi = 0 for i = 1, . . . , m. Hence u and v can be viewed as the positive and negative parts of a real vector w = u − v. Therefore eT (u + v) = ||w||1 and one can write z∗ = min{k1 ||d − w||∞ + k∞ ||w||1 }. w

 We can now state the robust equivalent for the new polyhedral uncertainty set. Theorem 2. The robust equivalent of the robust constraint a¯T x + (PT x)T ξ ≤ b, for all ξ ∈ Ξ = {ξ | B1 (0, k1 ) ∩ B∞ (0, k∞ )}, is the constraint in x and w a¯T x + k1 ||PT x − w||∞ + k∞ ||w||1 ≤ b. Proof. Using Lemma 3, we may write the robust equivalent as a¯T x + z∗ ≤ b. Since for any w one has z∗ ≤ k1 ||PT x − w||∞ + k∞ ||w||1 , we have that the new inequality

Robust Optimization for Environmental and Energy Planning

19

implies the robust inequality. By the strong linear duality theorem, the equivalence is achieved because there always exists a w∗ such that z∗ = k1 ||PT x−w∗ ||∞ +k∞ ||w∗ ||1 .  Remark 1. The ball B1 (0, k1 ) in Theorem 2 can be replaced by a ball B p (0, k p ) in the norm ` p , for any 1 < p < ∞. One obtains the following robust equivalent a¯T x + k p ||PT x − w||q + k∞ ||w||1 ≤ b with q such that 1/p + 1/q = 1. For implementation purposes, it is convenient to replace the norm expressions into linear inequalities. Proposition 1. The robust counterpart a¯T x + k1 ||PT x − w||∞ + k∞ ||w||1 ≤ b has the same set of solutions as the system of linear inequalities a¯T x + k1t + k∞ eT w ≤ b

(10a) T

w + te ≥ P x w + te ≥ −PT x w ≥ 0, t ≥ 0.

(10b) (10c)

Proof. Let t be a scalar variable and write the minimization problem yielding z∗ = minw k1 ||PT x − w||∞ + k∞ ||w||1 as the linear program min k1t + k∞ eT (u + v) ≤ b −te ≤ PT x + u − v ≤ te u ≥ 0, v ≥ 0, t ≥ 0. This program can be simplified as follows. Consider first the case (PT x)i > 0. The minimization operation entails ui = 0, and vi = max{(PT x)i − t, 0}. A similar reasoning yields vi = 0 and ui = max{(−PT x)i − t, 0}, when (PT x)i ≤ 0. Define w ≥ 0 by wi = vi if (PT x)i > 0 and wi = ui if (PT x)i ≤ 0. By construction, w is such that w + te ≥ PT x and w + te ≥ −PT x. Conversely, a solution (w,t) to (10) generates a solution (u, v,t) of the initial system. Indeed, set ui = 0 and vi = wi if (PT x)i > 0 and ui = wi and vi = 0 if (PT x)i ≤ 0. We have thus that the minimization problem is equivalent to min k1t + k∞ eT w w + te ≥ PT x w + te ≥ −PT x

20

F. Babonneau, J.-P. Vial, and R. Apparigliato

w ≥ 0, t ≥ 0. 

This concludes the proof.

4.3 Robust equivalent: bounded polyhedral uncertainty set The balls in `1 and `∞ are special cases of polyhedra. We extend the construction of the robust counterpart to the case of uncertainty sets defined by a bounded nonempty polyhedral set defined by {ξ | Qξ ≤ t}. In that case the robustness test for the constraint (a¯ + Pξ )T x ≤ b, ∀ξ ∈ Ξ is given by max{(PT x)T ξ | Qξ ≤ t} ≤ −a¯T x + b. ξ

Proposition 2. The condition max{(PT x)T ξ | Qξ ≤ t} ≤ −a¯T x + b ξ

is satisfied if and only if there exists a solution (x, u) for the linear system a¯T x + t T u ≤ b PT x − QT u = 0 u ≥ 0. Proof. In the proof, x is a fixed parameter. A direct derivation is obtained using the theory of duality in linear programming. To define the worst case in the robust constraint, we consider the linear programming problem max{(PT x)T ξ | Qξ ≤ t}. ξ

This problem is feasible and bounded. It has an optimal solution and its dual min{t T u | QT u − PT x = 0, u ≥ 0} u

also has an optimal value, which is equal to the optimal primal value. For all feasible primal-dual pair (ξ , u) we have the implication Qξ ≤ t T Q u − PT x = 0, u ≥ 0



⇒ (PT x)T ξ ≤ t T u,

with equality if and only if the pair (ξ , u) is optimal. If we substitute the bounding value t T u to the maximization in the left-hand side of

Robust Optimization for Environmental and Energy Planning

21

a¯T x + max{(PT x)T ξ | Qξ ≤ t} ≤ b, ξ

we obtain a stronger inequality, which coincides with the initial one when u is dual optimal. Consequently, the system a¯T x + t T u ≤ b PT x − QT u = 0 u ≥ 0, which contains the feasibility constraints of the dual is equivalent to the initial robust constraint. 

4.4 Uncertainties in the objective function Let us consider the following linear problem with uncertainty data n

min x

∑ c j (ξ )x j

j=1

Ax ≤ b. There is a big difference difference between the objective and the constraints. In the latter, the values achieved by the left-hand side are irrelevant, insofar as they are less than the right-hand side. In particular, the average value is not relevant. In contrast, the average value of the objective function value is by all means an important factor to be considered. Let us assume that the uncertain factor has a symmetric distribution around 0 and that c(ξ ) is linear. The average value of the objective is thus ∑nj=1 c j (0)x j and one can simply minimize it. But in so doing, one completely ignore possible bad outcomes with high values for the objective function. A possible alternative is to focus on the worst case and consider the problem min z

(11a)

x,z

n

∑ c j (ξ )x j − z ≤ 0, ∀ξ ∈ Ξ

(11b)

Ax ≤ b.

(11c)

j=1

The z value in (11) is an upper bound for the objective function for all possible realizations in the uncertainty set. Minimizing this worst case performance is an acceptable decision criterion, but one may argue that shifting from one problem to

22

F. Babonneau, J.-P. Vial, and R. Apparigliato

the other amounts to pass from one extreme (no concern for the risk) to another (no concern for the average performance). At minimal computational cost, one may consider a bi-criteria problem involving the two objectives. The Pareto frontier is obtained by solving the parametric problem n

min x

∑ c j (0)x∗j

j=1 n

∑ c j (ξ )x j ≤ z∗ + γ, ∀ξ ∈ Ξ

j=1

Ax ≤ b. In that formulation γ > 0 determines an arbitrage between the average performance and the guaranteed worst case performance.

4.5 Two-sided inequality constraints In the robust optimization paradigm, the constraints are immunized separately with respect to their specific uncertainty sets. In the case of the two-sided constraint n

b≤

∑ a˜ j x j ≤ b¯

j=1

the robust formulation involves the two constraints b ≤ a¯T x + (PT x)T ξ , ∀ξ ∈ Ξ and ¯ ∀ξ ∈ Ξ . a¯T x + (PT x)T ξ ≤ b, Assume the polyhedral uncertainty set Ξ = {ξ | B1 (0, k1 ) ∩ B∞ (0, k∞ )}. In view of Theorem 2, the two robust constraints generate their own robust counterpart b ≤ a¯T x − k1 ||PT x − v||∞ − k∞ ||v||1

(12)

¯ a¯T x + k1 ||PT x − w||∞ + k∞ ||w||1 ≤ b.

(13)

In view of Proposition 1, those two inequalities boil down into independent sets of inequalities. The point of this section is that the two sets of extra inequalities don’t need to be independent. Actually the same set is valid for both, thus saving significantly on the problem size.

Robust Optimization for Environmental and Energy Planning

23

To see this, it suffices to notice that one need not to choose different vectors v and w in the robust counterparts (12) and (13). Indeed, for a fixed x, the constraints are equivalent to k1 ||PT x − v||∞ + k∞ ||v||1 ≤ aT x − b k1 ||PT x − w||∞ + k∞ ||w||1 ≤ b¯ − aT x. To check satisfiability of these constraints, it suffices to replace v and w by the value that minimizes the left-hand sides. This can be conveniently formalized into a theorem similar to Proposition 1. Proposition 3. The robust counterpart of the robust two-sided constraint ¯ for all ξ ∈ Ξ = {ξ | B1 (0, k1 ) ∩ B∞ (0, k∞ )}, b ≤ a¯T x + (PT x)T ξ ≤ b, is k1 ||PT x − w||∞ + k∞ ||w||1 ≤ min{aT x − b, b¯ − aT x}. It can be represented by the system of linear inequalities k1t + k∞ eT w ≤ aT x − b k1t + k∞ eT w ≤ b¯ − aT x

(14a) (14b)

w + te ≥ PT x

(14c)

T

(14d)

w + te ≥ −P x w ≥ 0,

t ≥ 0.

The practical importance of this theorem is that the robust counterpart of a two-sided inequality is a set of inequalities with only one more inequality than in the case of a simple one-sided inequality. A similar treatment applies to a two-sided inequality with ellipsoidal uncertainty. We then would get the robust counterpart k||PT x||2 ≤ min{aT x − b, b¯ − aT x}.

4.6 Equality constraints The meaning of an equality constraint with uncertain coefficients is questionable. Using our standard representation of the uncertain parameters, we can formulate the robust equality constraint a(ξ ˜ )T x = b, with a(ξ ˜ ) = a¯ + Pξ and ξ ∈ Ξ . The uncertainty sets that have been considered so far have a non-empty interior and 0 ∈ Ξ . Consequently a˜T x = a¯T x + (PT x)T ξ = b for all ξ ∈ Ξ if and only if (PT x)T ξ ≡ 0

24

F. Babonneau, J.-P. Vial, and R. Apparigliato

for all ξ ∈ Ξ . This implies PT x = 0. In other words, the vector x should lie in the null space of PT . This is conveniently summarized in a proposition. Proposition 4. Suppose the uncertainty set Ξ has a non-empty interior. The robust equality constraint a¯T x + (Pξ )T x = b, ∀ξ ∈ Ξ is equivalent to the system of equations a¯T x = b and PT x = 0. The condition on x looks very restrictive. A possibility is that the presence of uncertain coefficients in an equality constraint signals an error in passing from a deterministic version of the problem to an uncertain one. For instance, some inequality constraints in the deterministic version are known to be necessarily tight at the optimum. This implicit knowledge makes it possible to write them as equalities in the deterministic formulation, but the true inequality formulation should be used in the uncertainty case. A more difficult situation occurs in multistage problems in which some decisions are recourses that can be adjusted once the uncertainty is revealed. We shall see an illustration of this problem in our energy environmental planning example in a later section dealing with robust optimization in the dynamic case.

5 Dynamic problem with recourse Multistage problems under uncertainty introduce a quantum of difficulties in that the successive decisions can be made on the basis of the information revealed so far. Those decisions will be named recourses thereafter, to emphasize the relation with the revealed information. Recourses are functions defined on richer and richer spaces. To cope with this extreme difficulty, it is reasonable to restrict the recourses to functions in a limited class, a class simple enough to allow the formulation of robust constraints, but rich enough to capture a meaningful part of the recourse possibilities. The proposed class is the set of linear functions of the revealed values of the uncertain factor ξ . Recourses in this class will be named Linear Decision Rules, LDR in short. This concept has been introduced for control problem [31] and for stochastic programming [29, 23]. It is used in Robust Optimization under the name of Affinely Adjustable Robust Counterpart (AARC) [8].

Robust Optimization for Environmental and Energy Planning

25

5.1 Linear decision rules Consider a typical constraint in a two-stage problem: a1 (ξ1 )T x1 + a2 (ξ1 , ξ2 )T x2 ≤ b(ξ1 , ξ2 ). We assume that x1 and x2 are variables with dimension, respectively, n and m. The coefficients a1 and a2 have also, respectively, dimension n and m. The dependence of coefficients a1 , a2 and b with the uncertainty underlying (ξ1 , ξ2 ) is given by the functions a1 (ξ1 ) = a¯1 + P1 ξ1

(15a)

a2 (ξ1 , ξ2 ) = a¯2 + P21 ξ1 + P22 ξ2 b(ξ1 , ξ2 ) = b¯ + bT1 ξ1 + bT2 ξ2 .

(15b) (15c)

In the last equality, b1 and b2 have same dimension than ξ1 and ξ2 . To capture the adaptive property of the recourse in stage 2, the variable x2 is replaced by a LDR, that is by a linear (more accurately, an affine) function of ξ1 , that is: x2 (ξ1 ) = x¯2 + Dξ1 . In the above expression, the components of x¯2 and D are the new decision variables in the formulation of the problem with uncertain parameters. Note that those variables are to be determined prior to knowing the value of ξ1 , but the actual value of the recourse x2 (ξ1 ) will be determined after the value of ξ1 has become known. The robust equivalent of the constraint of the two-stage problem is then (a¯1 + P1 ξ1 )T x1 + (a¯2 + P21 ξ1 + P22 ξ2 )T (x¯2 + Dξ1 ) −(b¯ + b1 ξ1 + b2 ξ2 ) ≤ 0, ∀(ξ1 , ξ2 ) ∈ Ξ . To build the robust counterpart, one must solve the optimization problem T T max (P1T x1 + P21 x¯2 + DT a¯2 − b1 )ξ1 + (P22 x¯2 − b2 )T ξ2

(16)

(ξ1 ,ξ2 )∈Ξ

+(P21 ξ1 + P22 ξ2 )T Dξ1 in which the variables x1 , x¯2 and D are parameters. We notice that this problem is quadratic with second order term (P21 ξ1 + P22 ξ2 )T Dξ1 . Remark 2. It is worth noticing that a LDR introduces uncertainty of its own. A constraint with deterministic coefficients, e.g., with P1 = P21 = 0, P22 = 0, b1 = 0 and b2 = 0, becomes uncertain because of the term (DT a¯2 − b1 )ξ1 .

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F. Babonneau, J.-P. Vial, and R. Apparigliato

5.2 Problems with uncertain recourse parameters If the coefficient a2 (ξ1 , ξ2 ) of the recourse variable is not fixed, the matrices P21 and P22 are not identically zero. Since D is not restricted to be in a certain class, the quadratic term in the objective of (16) can be indefinite. The maximization problem in (16) becomes non-convex and one cannot resort to duality theory to build a simple robust counterpart, as it has be done until now. There is an exception to this bad situation. If the uncertainty set Ξ is an ellipsoid, it can be shown that the problem (16) is equivalent to a convex problem on the cone of the positive semi-definite matrices. It becomes possible to derive a robust counterpart. We refer the reader to [11] and [8] for more details. We just mention that there exist efficient codes, e.g., the open source Sedumi [38], that can solve very efficiently problems with constraints on the set of positive semi-definite matrices.

5.3 Problems with fixed recourse To remain in the realm linear programming, we introduce the following assumption. Assumption 2 The a2 (ξ ) vector associated to the recourse is fixed (a2 (ξ ) ≡ a2 ). Proposition 5. Assume the dynamic problem has fixed recourse. The robust counterpart of a dynamic constraint with LDR and a polyhedral uncertainty set is given by a set of linear constraints. Proof. From the fixed-recourse assumption, we have P21 ξ1 + P22 ξ2 ≡ 0. Since the quadratic term of the objective disappears, we can apply Proposition 2 and express the robust counterpart as a set of linear constraints.  The fixed-recourse assumption confines the uncertainty to the first stage coefficients and to the right-hand sides. This is obviously limitative, but still relevant to interesting applications problems, for instance in supply chain management problem [6] or on the management of a hydraulic valley with uncertainty on the water supply [3, 2]. In some dynamic problems, we encounter constraints aT x = b, with a deterministic and b uncertain. We shall get an example of it in our case problem. There, aT x is a production that is achieved by combining factors in proportion given by x and b is a demand. The only way to match an uncertain demand is to make the x vector uncertain and have it adjusted to the random behavior of b. This is achieved by a LDR incorporating the random factor in b. Two options are open. The first one is to keep working with the equality constraint and achieve global robustness as described in subsection 4.6. This approach has been followed in [35]. An alternative

Robust Optimization for Environmental and Energy Planning

27

consists in relaxing the equality into the inequality aT x ≥ b, guaranteeing that the demand will be satisfied at the possible cost of disposal if the production exceeds the demand. This approach has been followed in [6].

6 Case study: uncertain demand with fixed recourse In this section, the case study deals with the model with uncertain demand and no uncertainty in the pollutant transfer coefficients. This is a multistage problem with fixed recourse for which the LDR methodology of the previous section is appropriate. To contrast Robust Optimization with more classical approaches, we have implemented a simple Stochastic Programming version. We built a discrete distribution to approximate the stochastic demand process used in the validation process. The number of branches is such that the so-called deterministic equivalent, i.e., the extensive formulation of the problem relatively to the event tree, has about the same size as the robust counterpart.

6.1 LDR and relaxed demand constraints We reformulate Problem (1) to eliminate equality constraints, either by simple elimination (constraints (1e)) or by relaxation (constraints (1d)). We get the following relaxed version of (1) min

x≥0,y≥0

∑(Or ∑ x p,t,r + Ir ∑ y p,t,r + Mr ∑(res p,t,r + ∑ y p,τ,r )) r

p,t

p,t

x p,t,r ≤ res p,t,r + ∑ y p,τ,r

p,t

(17a)

τ≤t

∀t ∈ T , p ∈ P, r ∈ R

(17b)

τ≤t

∑ x p,t,r ≥ dt

∀t ∈ T

(17c)

t,r

∑ ∑(E p,ρ x p,t,ρ )Gρ,r ≤ Q

∀t ∈ T , r ∈ R

(17d)

ρ∈R p

This more compact formulation has 72 variables and 51 constraints. We now define, for all pairs (p ∈ P, r ∈ R), the following linear decision rules for the production variables x p,t,r and the investment variables y p,t,r . 0 1 η1 x p,1,r = α p,1,r + α p,1,r

x p,2,r = x p,3,r =

1 2 0 η2 α p,2,r + α p,2,r η1 + α p,2,r 2 3 0 1 α p,3,r + α p,3,r η1 + α p,3,r η2 + α p,3,r η3

(18a) (18b) (18c)

28

F. Babonneau, J.-P. Vial, and R. Apparigliato 0 y p,1,r = β p,1,r

y p,2,r = y p,3,r =

(18d)

0 1 β p,2,r + β p,2,r η1 0 1 2 β p,3,r + β p,3,r η1 + β p,3,r η2 .

(18e) (18f)

In that definition the random variables ηi are the ones of the demand uncertainty defined in (3). Thus the variables x and y are function of the demand uncertainty. Note that the investment in t is decided before the demand at t is known, while the production is set after, hence the difference in the decision rules. If we replace the variables in the constraints of (17) by their linear decision rules, we obtain the LDR formulation min α,β ,ξ

0 0 0 + Ir ∑ β p,t,r + Mr ∑(res p,t,r + (4 − p)β p,t,r ) ∑(Or ∑ α p,t,r

(19a)

1 0 0 α p,1,r η1 + α p,1,r − β p,1,r ≤ res p,1,r

(19b)

r

p,t

p,t

p,t

∀p, r

2 1 1 α p,2,r η2 + (α p,2,r − β p,2,r )η1 + 0 0 0 +α p,2,r − β p,1,r − β p,2,r ≤ res p,2,r ∀p, r 3 2 2 1 1 1 α p,3,r η3 + (α p,3,r − β p,3,r )η2 + (α p,3,r − β p,2,r − β p,3,r )η1 + 0 0 0 0 +α p,3,r − β p,1,r − β p,2,r − β p,3,r ≤ res p,3,r ∀p, r

(∑

1 α p,1,r − dˆ1 )η1 +

p,r



0 α p,1,r

≥ d¯1

(19c)

(19d) (19e)

p,r

2 1 0 (∑ α p,2,r − dˆ2 )η2 + (∑ α p,2,r − dˆ1 )η1 + ∑ α p,2,r ≥ d¯2 p,r

p,r

(19f)

p,r

3 2 (∑ α p,3,r − dˆ3 )η3 + (∑ α p,3,r − dˆ2 )η2 + p,r

p,r

1 0 +(∑ α p,3,r − dˆ1 )η1 + ∑ α p,3,r ≥ d¯3 p,r

1 0 )Gi,r η1 + ∑ ∑(E p,i α p,1,r )Gi,r ≤ Q ∑ ∑(E p,i α p,1,r

i∈R p

(19g)

p,r

∀r

(19h)

i∈R p

2 1 )Gi,r η2 + ∑ ∑(E p,i α p,2,r )Gi,r η1 + ∑ ∑(E p,i α p,2,r

i∈R p

i∈R p

0 + ∑ ∑(E p,i α p,2,r )Gi,r ≤ Q

∀r

(19i)

∀r

(19j)

i∈R p

3 2 )Gi,r η3 + ∑ ∑(E p,i α p,3,r )Gi,r η2 + ∑ ∑(E p,i α p,3,r

i∈R p

i∈R p 1 + (E p,i α p,3,r )Gi,r η1 + i∈R p i∈R p 1 0 α p,1,r + α p,1,r η1 ≥ 0 ∀p, r

∑∑

0 )Gi,r ≤ Q ∑ ∑(E p,i α p,3,r

2 0 1 η1 + α p,2,r η2 ≥ 0 ∀p, r α p,2,r + α p,2,r 2 3 0 1 η1 + α p,3,r η2 + α p,3,r η3 ≥ 0 α p,3,r + α p,3,r

(19k) (19l) ∀p, r

(19m)

Robust Optimization for Environmental and Energy Planning 0 β p,1,r ≥0

29

∀p, r

(19n)

0 1 β p,2,r + β p,2,r η1 ≥ 0 ∀p, r 0 1 2 β p,3,r + β p,3,r η1 + β p,3,r η2 ≥ 0

(19o) ∀p, r.

(19p)

In that formulation all constraints depend on the random variables ξ and must be given a robust counterpart. For√each constraint we choose the uncertainty set to be the intersection of the B1 (0, k m) and B∞ (0, k) balls, where m is the number of random variables in the constraint. We also select different immunization factors k according to the type of constraints. Namely, we use kcap for the capacity constraints (19b)–(19d), kdem for the demand constraints (19e)–(19g) and kemi for the air quality constraints (19h)–(19j). In the experiments, we set kcap = kemi = 1 and we test different values for kdem . The robust equivalent of (19) has 419 variables and 502 constraints. It corresponds to an increase with a multiplicative factor 4 of size of the deterministic version (17). We report the results in Table 13. Deterministic

Predicted cost performance Observed cost performance

LDR with robust kdem = 0.4 kdem = 0.6 kdem = 0.7

Objective function 162.06 160.87

165.84 165.07

167.09 166.14

167.40 167.47

Constraints on the demand Scenarios with demand violation(s) in % 62.0 Conditional average relative violation in % 2.5 Average number of violations per scenario 2.0

20.4 0.8 1

9.9 0.6 1

0 -

Constraints on the air quality Total number of violated air quality constraints 0

53

39

82

Table 13 Variable demand: behavior on the sample of 1000 scenarios of a robust solution with LDR.

The conclusions to be derived from Table 13 are twofold. If we increase the immunization level kdem , the robust solution better and better tracks the (relaxed) demand constraint. On the other hand, a larger immunization factor induces more variability in the LDR and increases risks of violations in the (otherwise deterministic) air quality constraints. Note also the degradation of the cost performance when the immunization increases.

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F. Babonneau, J.-P. Vial, and R. Apparigliato

6.2 LDR and exact demand constraints In the previous section, we transformed the equality demand constraint in the deterministic model (1d) into the inequality constraint (17c) to match the uncertainty in the right-hand side. We now follow the approach of Section 4.6 and keep the equality in the demand constraint. In view of Proposition 4, the linear decision rule coefficients must satisfy the following conditions. Namely, we replace the demand equations (19e)–(19g) by the following set of equations 1 0 = dˆ1 , = d¯1 , ∑ α p,1,r ∑ α p,1,r p,r

(20a)

p,r

0 1 2 = d¯2 , ∑ α p,2,r = dˆ1 , ∑ α p,2,r = dˆ2 , ∑ α p,2,r p,r

p,r

(20b)

p,r

0 1 2 3 = d¯3 , ∑ α p,3,r = dˆ1 , ∑ α p,3,r = dˆ2 , ∑ α p,3,r = dˆ3 . ∑ α p,3,r p,r

p,r

p,r

(20c)

p,r

These conditions ensure that the demand constraints will be globally satisfied for all possible demands in the 3-dimensional space (R3 ). The model has now 412 variables and 487 constraints. In the numerical example, we keep replacing the other constraints (the capacity constraints (19b)–(19d), the air quality constraints (19h)–(19j) and the non-negativity constraints on the production and capacities) with their equivalent robust counterpart. To make the results comparable, we set kcap = kemi = 1 as before. The last set of constraints deals with non-negativity. To enforce those constraints with maximum chances we use an immunization level k equal to 1. The results are displayed in Table 14. Surprisingly enough, the results are very much alike the relaxed case Predicted cost performance Observed cost performance Scenarios with demand violation(s) in % Total number of violated air quality constraints

167.40 167.47 0 82

Table 14 Variable demand: behavior on the sample of 1000 scenarios of a globally robust solution with respect to the demand constraint.

with kdem = 0.7 in Table 13. This reflect the fact that it is relatively easy to track the demand by simple adjustment in the current period.

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6.3 Stochastic programming To apply stochastic programming, we model the demand process by a finite event tree. To end up with a problem with a size comparable to (19), we use a tree with 3 branches at each node and thus 27 scenarios in total. We assume that the probability factors η1 , η2 and η3 are i.i.d. with a uniform distribution. The best approximation of the uniform distribution by a finite distribution with 3 elements consists in partitioning the range space [−1, 1] into 3 elements and choose the mid-point value of each subinterval to be the representative of the subinterval. The probability for each representative is 31 . The tree is represented in Figure 3. If we start from the compact problem (17), the stochastic formulation has 624 variables and 663 constraints (419 variables and 502 constraints for the LDR formulation). Deterministic Stochastic Predicted cost performance Observed cost performance Scenarios with demand violation(s) in % Conditional average relative violation in % Average number of violations per scenario

162.06 160.87 62.0 2.5 2.0

165.18 164.80 50.4 1.0 1.4

Table 15 Variable demand: behavior on the sample of 1000 scenarios of a stochastic programming solution built on an approximating event tree with 3 branches per node.

Table 15 suggests that the model cannot really cope with the uncertainty in the demand constraints. Indeed, production is determined relatively to the representative demand at the node. In the simulations, the demand is never equal to the value assigned to the node. With the symmetric distribution it lies above this value in 50% of the cases. However, we observe that the stochastic programming solution improves upon the deterministic solution, and contrary to the robust solution with LDR, it never induces air quality constraints. This is so because productions at each node are chosen to meet those constraints. In the simulations, the productions are either kept unchanged or lowered to match lower demand; so, no air quality constraint violation is to be expected. It would be preposterous to derive definitive conclusions from this short example, but we can stress few facts. Stochastic Programming is at pain to meet constraints, while Robust Optimization controls violations pretty well. Stochastic programming treats constraints at privileged points: the nodes on the event tree, but Robust Optimization deals with the continuum of events in the event tree. To improve the control on the constraint in Robust Optimization, it suffices to increase the immunization level. A similar objective of better constraint satisfaction with Stochastic Programming calls for a larger event tree to better approximate the stochastic process, an approach that quickly leads to an numerically intractable optimization problem.

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F. Babonneau, J.-P. Vial, and R. Apparigliato t1

yt1r

t3

t2 xt1r yt2r

xt12r yt3r

xt3r d3 = 12.80 13.27

d2 = 11.26 11.66

13.74 13.20 13.67 14.14

d1 = 9.66 12.06

13.60 14.07 14.54 13.13 13.60

11.60 d1 = 10.00

12.00

14.07 13.53 14.00 14.47

12.40

13.93 14.40 14.87 13.46 13.93

d1 = 10.33

11.93 12.33

14.39 13.86 14.33 14.79

12.73

14.26 14.73 15.19

Fig. 3 Scenario decision tree

On the other side, Stochastic Programming delivers a richer output, provided one has reliable and tractable information on the probability distribution. This takes the form of the objective performance on each branch on the event tree, that gives an interesting approximation of the probability distribution of the optimal function value. It is even possible to add constraints, such as CVaR (conditional value at risk), to take into account the amount of risk.

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7 Case study: uncertainty in demands and pollutant transfers We consider now the more involved case where both the demands and the pollutant transfer coefficients are uncertain. As it has been pointed out earlier, LDR introduce an uncertainty of their own in the constraints in which they appear. If some of these constraints have themselves uncertain coefficients, the product of an uncertain coefficient with the uncertain decision variable (defined by the LDR) generates a bilinear form in the uncertainty. One cannot anymore use duality as in Section 4 to exhibit the robust counterpart. To cope with this difficulty, we propose two empirical approaches, one that fully remains in the realm of Robust Optimization and a hybrid one that mixes Robust Optimization and Stochastic Programming.

7.1 A fully robust optimization approach In the fully robust optimization approach, we eliminate the LDR in the air quality constraints by the following gimmick. We introduce an artificial variable to bound the value produced by the LDR. In our problem of interest the artificial variable represents a deterministic upper bound on the productions prescribed by the LDR. In view of the nature of air quality constraints with nonnegative coefficients only, higher productions entail higher emissions and, after pollutant transmission, deteriorate the local air quality. Therefore, it makes sense to replace the LDR value by an upper bound to comply with the worst case requirement. This idea is implemented as follows. The air quality constraints (19h)–(19j) in the LDR formulation (19) are replaced by (E p,i x p,t,r )Gi,r ≤ Q ∀r,t i∈R p 0 1 α p,1,r + α p,1,r η1 ≤ x p,1,r ∀p, r 0 1 2 α p,2,r + α p,2,r η1 + α p,2,r η2 ≤ x p,2,r ∀p, r 0 1 2 3 α p,3,r + α p,3,r η1 + α p,3,r η2 + α p,3,r η3 ≤ x p,1,r

(21a)

∑∑

(21b) (21c) ∀p, r.

(21d)

where the matrix G is uncertain. Because the upper bound x is deterministic, we now can deal with uncertain coefficients G in (21a) and write their robust counterpart. Note that constraints (21b)–(21d) together with the non-negativity constraints (19k)–(19m) form a group of two-sided constraints. We use Proposition 3 to cut down the number of constraints and variables in the robust counterpart of those two-sided constraints. Note that the production LDR variables are replaced by an

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F. Babonneau, J.-P. Vial, and R. Apparigliato

upper bound in the air quality constraints (21a), but not in the capacity constraints (19b)–(19d). The formulation has 487 variables and 542 constraints The numerical experiments aim to compare the LDR solution computed in subsection 6.1 and the solution of the fully robust optimization approach. In the latter, we used different immunization levels kemi for the air quality constraints. In all cases we used the same immunization level kdem = 0.7 for the demand constraints. The results are displayed in Table 16. Standard LDR

Predicted cost performance Observed cost performance

Objective function 167.40 167.47

LDR with robust const. kemi = 0 kemi = 0.5 kemi = 1 167.66 167.51

169.21 169.28

170.72 170.78

Constraints on the demand Scenarios with demand violation(s) in % 0 Conditional average relative violation in % Average number of violations per scenario -

0 -

0 -

0 -

Constraints on the air quality Total number of violated air quality constraints 829

802

150

11

Table 16 Variable demand and variable transfer coefficients: behavior on the sample of 1000 scenarios of a robust solution with LDR.

The reader will notice that the new approach with an the immunization level in the air quality constraint set to kemi = 0, yields a solution such that the number of violated air quality constraints is slightly less (802 instead of 829) than with the standard LDR approach, while the cost performance is slightly worse (167.66 instead of 167.40). In both cases, the model treats the air quality constraint in a deterministic way, but in the new approach the LDR production variable is replaced by an upper bound in the air quality constraint. This more conservative approach is a safer for the constraint but also more costly. However the difference almost negligible. If we increase the immunization level kemi , the solution with the new approach ensures better and better protection against violations of the air quality constraints.

7.2 A hybrid approach: stochastic programming with robust constraints We now assume both the coefficients of the source-receptor matrix G and the demands are uncertain. As in the previous section, the demands are defined by (3). Because we cannot use LDR in a constraint with other uncertain coefficients, we

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35

choose to represent the demand uncertainty via a finite event tree. We shall use stochastic programming to define the decision to be taken at each node of the tree. The new feature is that those decisions will be requested to be robust with respect to the air quality constraints. Therefore, we shall have to introduce different robust counterparts of those constraints at each node of the tree. As before, we use the event tree displayed in Figure 3. The hybrid formulation with a demand event tree and robust air quality constraints at the tree nodes has 819 variables and 819 constraints. The results are displayed in Table 17. Deterministic

Predicted cost performance Observed cost performance

Objective function 165.18 164.80

Hybrid k = 0.5 k = 1.0 k = 1.2 166.35 167.46 167.88 165.98 167.09 167.51

Constraints on the demand Scenarios with demand violation(s) in % 50.4 Conditional average relative violation in % 1.0 Average number of violations per scenario 1.40

50.4 1.0 1.36

50.4 1.0 1.36

50.4 1.0 1.36

Constraints on the air quality % of simulations with air quality violations 54.6 Average number of air quality violations 1.7

27.1 1.5

1.5 1.2

0 0

Table 17 Variable demand and variable transfer coefficients: behavior on the sample of 1000 scenarios of a solution obtained by the mixed approach.

If we compare the results of the LDR solution in Table 14 with those of the hybrid approach in Table 17, we observe that the latter can achieve much better control of the air quality constraints (even though there was no exogenous uncertainty in the first case and some in the other), while the former enables full control of the demand constraint, something that stochastic programming cannot achieve.

8 Probability of constraint satisfaction Prior to applying the robust optimization to the example, it is worth relating robust optimization to chance-constrained programming. This alternative approach, that was introduced in the late fifties by Charnes and Cooper [22], replaces the deterministic constraint by a constraint in probability. Namely, it considers that a solution is admissible if the probability that the constraint be satisfied with this solution is higher than a certain threshold. Since robust optimization is not built on probability grounds, comparing the two approaches seems to be out of order. Surprisingly enough, it turns out that one can assess bounds on the probability of satisfaction of

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F. Babonneau, J.-P. Vial, and R. Apparigliato

a constraint by a robust solution, at the cost of mild assumptions on the probability distributions of the uncertain parameters. The result stems from a theorem to be found in Chapter 3 of [5]. We give here a slightly stronger version than the one to be found in the literature.

8.1 Bounds on the probability of constraint satisfaction Theorem 3. Let ξi , i = 1, . . . , m be independent random variables with values in interval [−1, 1] and with average zero: E(ξi ) = 0. If zi , i = 1, . . . , m are deterministic coefficients, we have for all k ≥ 0 s ) ( m m k2 ). Prob ξ | ∑ zi ξi > k ∑ z2i ≤ exp(− 1.5 i=1 i=1 Remark 3. The assumption in Theorem 3 deals with the support of the random variables, their expectation and their independence. No other probabilistic assumption is made, in particular, nothing concerning distributions. The following two lemmas are used in the proof of Theorem 3. Lemma 4 (Chebytchev inequality). Prob(X ≥ a) ≤ e−a E(eX ). Proof. E(eX ) = E(eX |X < a)Prob(X < a) + E(eX | X ≥ a)Prob(X ≥ a) ≥ E(eX |X ≥ a)Prob(X ≥ a)

(because eX > 0 ⇒ E(eX | X < a) ≥ 0)

a

≥ e Prob(X ≥ a). The last inequality is from E(eX |X ≥ a) ≥ E(ea |X ≥ a) = ea .  t2

Lemma 5. The inequality e α + t ≥ et with 1 ≤ α ≤ Proof. The function et

2 /α

q

8 3

is valid for all t ≥ 0.

+ t − et is decreasing in α for all t > 0. To proveqthe property, we consider the extreme value of α. We assume in the following α = 83 . The derivatives of function g(t) = et

2 /α

+ t − et are

Robust Optimization for Environmental and Energy Planning

g0 (t) = and

37

2 t 2 /α te + 1 − et α

2 2 2 g (t) = ( t 2 + 1)et /α − et = et α α

00



 2 2 2 t 2 /α−t ( t + 1)e −1 . α α

(22)

p We note that g0 (0) = 0 and g00 (0) = 2/α − 1 ≥ 3/2 − 1 > 0. We show first that g00 (t) ≥ 0 for all t ≥ 0. Thus we show that the term in parenthesis in the right hand side of (22) is positive. From the inequality ex ≥ 1 + x, we have    2 2 2 2 2 2 t2 t 2 /α−t ( t + 1)e − 1 ≥ h(t) = t +1 1 + − t − 1. α α α α α The right component is a fourth degree polynomial that is convex because α < 4 and     12 4 12 4 2 2 2 00 t − 2t + 1 = 2 (1 − t) + ( − 1)t > 0. h (t) = 2 α α α α The polynomial 2 h (t) = α 0



 r 3 8 3 6 2 6 t − t + t −1 = α2 α α 2

r 3

3t − 3

! r 3 2 3 t +3 t −1 2 2

p is thus increasing; its value at 0 is − 3/2 and it tends to +∞ when t tends to +∞. It has a unique real root in t¯ = 0.366. The function has its minimum in t¯ and thus g00 (t) ≥ h(t) ≥ h(t¯) = 0.0208. We proved that g00 (t) ≥ 0 on the interval [0, 1] and thus that the function g0 (t) is increasing. Since g0 (0) = 0, we conclude that g0 (t) ≥ 0 and thus that g(t) is increasing. Finally since g(0) = 0, we have g(t) ≥ 0 on 0 ≤ t ≤ 1.  Proof. Now we can prove Theorem 3. This proof is derived from [5]. From Lemma 4 we write Prob(∑ zi ζi > k||z||) ≤ e−k||z|| E(e∑i zi ζi ). i

Since the variables ζi are independent, then E(e∑i zi ζi ) = E(∏i ezi ζi ) = ∏i E(ezi ζi ). We also have E(ezi ζi ) = 1 + ∑ E( k≥2

(zi ζi )k ) (because E(ζi ) = 0) k!

|zi |k k≥2 k!

≤ 1+ ∑

(because |zi ζi | ≤ |zi |) z2 i

≤ e|zi | − |zi | ≤ e α .

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F. Babonneau, J.-P. Vial, and R. Apparigliato

The last inequality comes from Lemma 5. Thus we can write z2 i

Prob(∑ zi ζi > k||z||) ≤ e−k||z|| ∏ e α = e−k||z||+

||z||2 α

.

i

i

As the relation Prob(∑ zi ζi > k||z||) = Prob(γ ∑ zi ζi > γk||z||) i

i

is true for all γ > 0 , then we have Prob(∑ zi ζi > k||z||) ≤ min e−γk||z||+γ

2 2 ||z|| α

γ>0

i

≤ e−

α 2 k2 4

k2

≤ e− 1.5 .

Remark 4. The standard result gives the bound exp(−k2 /2). An experimental study of the function exp(t 2 /α) + t − exp(t) shows that the maximum value of α, for which Lemma 5 remains true, is 1.79. The bound in the proposition is tighten to exp(−t 2 /1.25). Let us apply the result when the uncertainty set is the ball B2 (0, k) and the variable support for ξ j is the interval [−1, 1]. We consider the robust constraint a¯T x + (Pξ )T x ≤ 0, ∀ξ ∈ B2 (0, k). From Lemma 1 max(PT x)T ξ ≤ max{(PT x)T ξ | ||ξ ||2 ≤ k} = k||PT x||2 . ξ ∈Ξ

ξ

Letting z = PT x, Theorem 3 yields   k2 Prob (a¯ + Pξ )T x ≤ 0 | ξ ∈ Ξ ≥ Prob (PT x)T ξ ≤ k||PT x||2 ≥ 1 − exp(− ). 1.5 Note that for k = 2.63 we get the bound 0.99 on the probability. Using Theorem 3, we can also bound the probability associated with an uncertainty set defined as the intersection of two balls in the `1 and `∞ norms respectively. This done in the next corollary. Corollary 1. Let ξi , i = 1, . . . , m be independent random variables with values in the interval [−1, 1] and average zero: E(ξi ) = 0. If zi , i = 1, . . . , m are deterministic coefficients, we have for all k ≥ 0 ( ) m m k2 ). Prob ξ | ∑ zi ξi > k ∑ |zi | ≤ exp(− 1.5 m i=1 i=1

Robust Optimization for Environmental and Energy Planning

39

Proof. The proof results from the relation between the norms `1 and `2 in Rm that implies √ B2 (0, k/ m) ⊆ B1 (0, k). √  Replacing k by k/ m in Theorem 3 we get the result.

8.2 Uncertainty set versus constraint satisfaction To get a flavor of the implication of Theorem 3 and its corollary, it is worth comparing √ the volume of the certainty set Ξ = B∞ (0, 1) with the volumes of the balls B1 (0, k m) and B2 (0, k), for some k guaranteeing √ a large probability of satisfying the uncertain constraint. Since the ball B1 (0, k m) is a rotation of the ball B∞ (0, k), it has the same volume as the ball B∞ (0, k), and hence a volume larger √ than the ball m) intersects B∞ (0, 1) by a factor km (for k ≥ 1). Notwithstanding, the ball B (0, k 1 √ √ the ball B∞ (0, 1) for k ≤ m. Hence the uncertainty set Ξ = B∞ (0, 1) ∩ B1 (0, k m) is smaller than the certainty set B∞ (0, 1), but not in large proportion. The situation with the ball B2 (0, k) is dramatically different. This fact is discussed in [5] and is summarized by the inequality √  r m VolB2 (0, k) (k π)m eπ . = ≤ k VolB∞ (0, 1) 2mΓ (m/2 + 1) 2m For k = 2.63 and m = 30 the ratio on the right hand-side is strictly less than 1 and goes super-exponentially fast to zero as the dimension of the space of uncertainty factor goes to infinity. Hence in large dimension the uncertainty set may be incommensurably smaller than the certainty set, yet a solution which is robust with respect to this uncertainty set may achieve a very high probability of satisfying the constraint. At this point, it is worth wondering whether it is appropriate to rely on an intuitive justification of the robust optimization based on the idea that it is necessary to select a large uncertainty set to achieve a high probability of constraint satisfaction. For sure, the probability of satisfying the constraint is at least as large as the probability associated with the uncertainty set, but the reverse is not true. We had a first confirmation of this fact by comparing the volume of the B2 (0, k) uncertainty set with the volume of the certainty set B∞ (0, 1). We now provide an example that shows that on some circumstances, it is possible to achieve very high probability while the uncertainty set has very small probability, even a zero probability! Consider the constraint

m

∑ ai xi ≤ b

i=1

(23)

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F. Babonneau, J.-P. Vial, and R. Apparigliato

with uncertain coefficients ai = a¯i + aˆi ξi , where ξi ∈ Ω . (Note that it is a particular case of the model a = a¯ + Pξ where P is the diagonal matrix with main diagonal a.) ˆ We define an uncertainty set Ξ ⊂ Ω and work with the robust constraint m

∑ (a¯i + aˆi ξi )xi ≤ b, ∀ξ ∈ Ξ .

i=1

Suppose we can assess a probability distribution for ξ on Ω . We have the implication ∑m i=1 (a¯i + aˆi ξi )xi ≤ b, ∀ξ ∈ Ξ and Prob(ξ ∈ Ξ ) ≥ 1 − α ⇓ Prob(∑m i=1 (a¯i + aˆi ξi )xi ≤ b) ≥ 1 − α. Truly enough, the reverse implication does not hold. Yet, the direct implication suggests that choosing an uncertainty set with large probability might be a good strategy. The following example reveals that the strategy can be overly conservative. In this example we assume the ai are independent random variables  ai =

1 with probability 1/2 0 with probability 1/2.

(24)

This uncertain constraint is cast into a robust optimization framework as follows. For the time being, forget about the probability distribution and write the coefficients in the form ai = 12 + 12 ξi with −1 ≤ ξi ≤ 1. Let the uncertainty set be the ellipsoid Ξ = {ξ | ||ξ ||2 ≤ k}. The robust counterpart of the constraint 1 1 xi + ∑ xi ξi , ∀ξ ∈ Ξ 2∑ 2 i i is

kr 1 xi + xi2 ≤ b. ∑ 2 i 2 ∑ i

Since the ξi are independent random variables with zero mean with range [−1, 1], we can apply Theorem 3 and get s ( ) m m k2 Prob ξ˜ | ∑ zi ξ˜i > k ∑ z2i ≤ exp(− ) 1.5 i i=1 for arbitrary zi . For k = 2 we get the bound 0.07 on the probability of constraint violation. If we apply the theorem to the robust counterpart, we have that a solution to r 1 xi + ∑ xi2 ≤ b ∑ 2 i i

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41

has a probability 1 − 0.07 = 0.93 to satisfy the probabilistic constraint m

∑ ai (ξi )xi ≤ b

i=1

for any probability distribution satisfying the (weak) hypothesis of the theorem. In particular, this is true for (24). On the other hand, the set Ξ = {ξ | ||ξ ||2 ≤ 2} does not contain any single realization of the random variable ξ as soon as ξ has dimension larger than 4. In other words Prob(ξ ∈ Ξ ) = 0. So, we imposed x to be robust with respect to an uncertainty set2 having probability 0, but we still guarantee that the robust solution satisfies the uncertain constraint with a probability at least 0.93. Clearly, Robust Optimization does much more than the intuition suggests. Let us pursue the discussion with this example. Consider now the same version of the problem but with binary variables xi and b = 60. The robust counterpart (still with k = 2) r 1 x + i ∑ xi2 ≤ 60 2∑ i i is equivalent to a bound on the number variables that can set to the value 1. We easily find that this number is N = 100. We now ask the question: how good is the lower bound 0.93 on the true probability of constraint satisfaction? Without loss of generality, we assume that the robust solution is xi = 1, i = 1, . . . 100 and zero otherwise. The left-hand side in the constraint ∑m i=1 ai xi ≤ 60 is a binomial random variable, with parameter 1/2. From the tables, we get that the probability of satisfaction is 0.98. It is certainly an improvement upon 0.93, but still it is quite an achievement to get 0.93 with a robust approach, in view of the weak assumption on the true distribution in Theorem 3.

9 Extension: Globalized robust optimization As we have seen it again and again, robust optimization deals with worst cases with respect to uncertainty sets. Robust optimization concentrates on solutions that remain feasible for all realizations within the uncertainty sets, but is silent about realizations that lie outside. In particular it does not take into account the magnitude of the violation. Globalized robust optimization [4] proposes an extension that admits possible constraint violations, but control their magnitude. 2

Note that in this example, the ellipsoidal set has either probability 0, if k is small or probability one.

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9.1 The concept of globalized robust optimization A globalized robust solution must satisfy the following two criteria: 1. The solution is robust for all realizations in the uncertainty set. 2. If a realization falls outside the uncertainty set, a violation of the constraint is tolerated, but this violation must “remain under control”. The fuzzy concept of remaining under control must be clarified, but we already see that a globalized robust solution is robust. The concept is then more restrictive, but it covers all cases, inside and outside the uncertainty set. To make the definition operational, one needs to be more specific about the meaning of “being under control”. In [4] the authors suggest to consider the distance from a current realization to the uncertainty set. If this distance is positive, i.e., if the realization is strictly exterior to the uncertainty set, a violation is acceptable, but its magnitude should be less than a fixed multiple of this distance. This definition applies to the realizations within the uncertainty set, because, the distance is null, and no violation is permitted. To formalize the idea, we first define an arbitrary distance to the uncertainty set. As in the previous examples, we consider the constraint (a¯ + Pξ )T x = a¯T x + (PT x)T ξ ≤ b to be satisfied for any ξ in the uncertainty set. We assume that Ω = Rm and that Ξ is convex. We introduce the convex distance function d(ξ , Ξ ) between ξ and the set Ξ . A solution is called globally robust if it satisfies a¯T x + (PT x)T ξ ≤ b + αd(ξ , Ξ ), ∀ξ ∈ Ω where α > 0 is a user parameter. This parameter is chosen in function of the tolerated violation. In this formulation, ξ can be given any value3 in Rm . To check whether a solution x meets the globalized robustness requirements, it suffices to replace the uncertain terms by its maximal value a¯T x + max{(PT x)T ξ − αd(ξ , Ξ )} ≤ b. ξ ∈Ω

To make things more precise, let us specify that the distance function is generated by a norm. Let δ : Rm → R+ be this norm, and define the distance as d(ξ , Ξ ) = min δ (ξ − ξ 0 ). ξ 0 ∈Ξ

The maximization operation in the left-hand side of the globalized robust equivalent constraint is a convex problem (maximizing a concave function). It can be written 3

This requirement may be excessive and unrealistic. One could think that the set is bounded, possibly with large bounds. This possibility is shortly discussed at the end of this section.

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43

as max {(PT x)T ξ − α min δ (ξ − ξ 0 )} ξ 0 ∈Ξ

ξ ∈Rm

= max {(PT x)T ξ + α max {−δ (ξ − ξ 0 )}} ξ 0 ∈Ξ

ξ ∈Rm

= max{(PT x)T ξ − αδ (ξ − ξ 0 ) | ξ ∈ Rm , ξ 0 ∈ Ξ }. We have thus the globalized robust equivalent written as a¯T x + max{(PT x)T ξ − αδ (ξ − ξ 0 ) | ξ ∈ Rm , ξ 0 ∈ Ξ } ≤ b.

(25)

9.2 Globalized robustness with linear programming We shall use this expression with a choice of the norm that keeps the whole formulation in the realm of linear programming. To this end, we select a polyhedral uncertainty set and an appropriate norm that can be described by a finite number of linear inequalities, such as those generated by the balls in the `∞ and/or `1 norms. The distance to this uncertainty set will also be defined with respect to `∞ and/or `1 , so that everything can be translated into linear inequalities. In principle, the combination of the `∞ and `1 norms used in defining the distance may be totally independent of the structure of the uncertainty set, but we choose to analyze a case where the uncertainty set and the distance function derive both from the same norm. Consider the uncertainty set Ξ = {ξ ∈ Ω | ||ξ ||1 ≤ k1 , ||ξ ||∞ ≤ k∞ } defined by the intersection of the ball with radius k∞ in the norm √`∞ and the ball with radius k1 in the norm `1 . (In practice we often choose k1 = k∞ m.) This polyhedron can be used to defined a norm as follows. Consider the homothetic set Ξ (t) = {ξ ∈ Ω | ||ξ ||1 ≤ k1t, ||ξ ||∞ ≤ k∞t} with t > 0. We can use this polyhedron to define the norm δ (µ) = min{t | µ ∈ Ξ (t)} = max{ t>0

1 1 ||µ||1 , ||µ||∞ }. k1 k∞

The distance from a point ξ to the set Ξ , is now defined by the optimization problem d(ξ , Ξ ) = min{δ (ξ − ν) | ν ∈ Ξ }. ν

Figure 4 illustrates the distance from √ ξ to the set Ξ with the selected norm in the 2-dimensional space with k1 = k∞ 2. The blue dotted curves represent the sets of points that are at the same distance of Ξ . The point ν ∈ Ξ is one of the closest point

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from ξ . (The vector ν is not unique because the distance function is not strictly convex.) The computed distance is d(ξ , Ξ ) = δ (ν, ξ ) = 2.53.

d = 2.53 d=2 d=1 ξ

ν

Ξ

Fig. 4 Distance d(ξ , Ξ ) with the composite norm

Proposition 6. Let a˜T x ≤ b be a constraint with uncertain elements a˜ = a¯ + Pξ . Let Ξ = {ξ ∈ Ω | ||ξ ||1 ≤ k1 , ||ξ ||∞ ≤ k∞ } be the uncertainty set and let d(ξ , Ξ ) = minξ 0 ∈Ξ δ (ξ − ξ 0 ) be the distance of a point ξ to Ξ , where δ (µ) = mint>0 {t | µ ∈ Ξ (t)} = max{ k11 ||µ||1 , k1∞ ||µ||∞ } is the norm induced by the polyhedron Ξ . Finally, let α be the coefficient of globalized robustness. The globalized robust equivalent of the globalized robust constraint a¯T x + (PT x)T ξ ≤ b + αd(ξ , Ξ ), ∀ξ ∈ Ω is given by the two constraints a¯T x + k1 ||PT x − u||∞ + k∞ ||u||1 ≤ b T

k1 ||P x − v||∞ + k∞ ||v||1 ≤ α.

(26a) (26b)

Proof. Let us study the maximization problem yielding the worst case in the definition of the robust equivalent and show it to be a linear programming problem. First, let us introduce the auxiliary variable ζ ∈ R and write the maximization problem yielding the globalized robust equivalent as z∗ = max{(PT x)T ξ 0 − αζ | δ (ξ 0 − ξ ) ≤ ζ , ζ ≥ 0, ξ ∈ Ξ , ξ 0 ∈ Ω }. ξ ,ν,ζ

Note that we have added the redundant constraint ζ ≥ 0. The objective maximization ensures ζ to be as small as possible, enforcing the equality δ (ξ 0 − ξ ) = ζ . Introducing the new variable η = ξ 0 − ξ and recalling that the norm δ is the maximum of two norms, we obtain the alternative formulation

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z∗ = max (PT x)T (ξ + η) − αζ ξ ,η,ζ

||η||∞ ≤ k∞ ζ , ||η||1 ≤ k1 ζ , ζ ≥ 0 ||ξ ||∞ ≤ k∞ , ||ξ ||1 ≤ k1 .

For fixed ζ , the problem is separable in ξ and η. By Lemma 2, we have  z∗ = max min k1 ||PT x − u||∞ + k∞ ||u||1 + ζ (k1 ||PT x − v||∞ + k∞ ||v||1 − α) u,v ζ ≥0  = min max k1 ||PT x − u||∞ + k∞ ||u||1 + ζ (k1 ||PT x − v||∞ + k∞ ||v||1 − α) . u,v ζ ≥0

The interchange between the max and min operators is allowed by the strong duality theorem in linear programming. The last expression can be simplified upon noticing that the inner maximization is either unbounded if k1 ||PT x − v||∞ + k∞ ||v||1 > α, or takes the value k1 ||PT x − u||∞ + k∞ ||u||1 if k1 ||PT x − v||∞ + k∞ ||v||1 ≤ α (because ζ = 0 in that case). Thus z∗ = min{k1 ||PT x − u||∞ + k∞ ||u||1 | k1 ||PT x − v||∞ + k∞ ||v||1 ≤ α}. u,v

As shown in Proposition 1, the above linear programming problem can be explicitly recast in terms of a linear objective and linear inequality constraints. Moreover, if feasible, it achieves its optimal value. This makes it possible to write the deterministic equivalent as a¯T x + k1 ||PT x − u||∞ + k∞ ||u||1 ≤ b k1 ||PT x − v||∞ + k∞ ||v||1 ≤ α.  Remark 5. A theorem similar to Proposition 1 can be stated for the globalized robust equivalent (26) with norms l1 and l∞ . We leave it to the reader to perform the simple transformations that lead to equivalent linear inequalities. Remark 6. In the definition of the globalized robustness, we assumed that the set Ω of all possible realizations is Rm . This makes the globalized robustness condition rather demanding. It may be worth considering that Ω ⊂ Rm is constrained by box constraints, such as Ω = {ξ | ξ ≤ ξ ≤ ξ¯ }. In that case, the globalized robust equivalent calls for two additional nonnegative variables and the constraints a¯T x + k1 ||PT x − u − w + t||∞ + k∞ ||u||1 + ξ¯ T w − ξ T t ≤ b k1 ||PT x − v − w + t||∞ + k∞ ||v||1 ≤ α w ≥ 0, t ≥ 0.

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If we set w = 0 and t = 0, we retrieve (26), confirming that reducing the span of Ω enlarges the set of solutions. Let us add few words of comments. Because of the additional constraint (26b), the set of globalized robust solutions is more restricted than the set of standard robust solutions. The globalized robustness uses the violation tolerance factor α. The smaller this parameter, the smaller must be the constraint violation. On the contrary, if α tends to infinity, constraint (26b) becomes inactive and we retrieve the plain robust solution. In this presentation we used polyhedral uncertainty sets. Similar results can be obtained with ellipsoidal uncertainty sets. An interesting application to multi-echelon, multi-period inventory control is reported in [7].

9.3 Case study: Globalized robustness with uncertain demands We now apply the concept of globalized robust optimization to the LDR formulation described in Subsection 6.1. We remind the reader that in Subsection 6.1 only the demands were uncertain and that the experiments were performed with different values of kdem . Here we propose to test different values of α on the particular case kdem = 0.4. We report the simulation results for each globalized robust solution in Table 18. LDR solutions with kdem = 0.4 α = 0.3 α = 0.2 α = 0.1 Predicted cost performance Observed cost performance Scenarios with demand violation(s) in % Conditional average relative violation in % Average number of violations per scenario Total number of violated air quality constraints

165.84 166.26 165.07 165.71 20.4 16.0 0.8 0.6 1 1 69

69

166.82 166.62 1.1 0.4 1 69

Table 18 Variable demand: behavior on the sample of 1000 scenarios of a globalized robust solution with LDR.

We notice that for α = 0.3 we retrieve the standard robust solution. Lower values of α permit a better control of the demand violations. The model formulation has 433 constraints and 509 variables (419 variables and 502 constraints for the standard LDR formulation).

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10 Conclusion The main goal of this chapter was to present an alternative approach to the dealing with uncertainties in environmental and energy planning. To this end, we introduced basic concepts in robust optimization and we applied them to an illustrative example. For the sake of the exposition, we chose an overly simplified example of modest size with three periods only. In practice, the models that are commonly used in the area of environmental and energy planning are much larger and much more complex, with thousands of variables and constraints. One should expect that the robust optimization methodology generates robust counterparts of much larger size, possibly to the point that they become impractical. This view must be qualified, by differentiating the type of uncertainties that are taken into consideration. The main difficulty stems from the multi-period feature of the models with uncertainties described as stochastic process. At this point, there is a sharp distinction to be made, depending on whether the realizations of the stochastic process are progressively revealed to the decision-maker, or only at once at the end of the terminal stage. If the exact value of the process remains unknown till the horizon of the model, the decision process is essentially static: all present and future decisions are to be taken at once, but the eventual effect of uncertainty is revealed in a second phase with no possibility of recourse. We treated an example of this sort in Section 3. The fact that the size of robust counterpart increases linearly with the dimension of the uncertainty factors suggests that, in this situation, even large models could be handled by robust optimization. The situation is much more dramatic if the uncertainty is progressively revealed in time. Then, the decision process must be adapted to the revealed information, a fact that introduces a major, fundamental difficulty for all known methods, except in some special cases4 . Linear decision rules for problems with fixed recourses may provide an acceptable approximation of an adaptive behavior and yield interesting hindsight, as those obtained in Section 6. We even suggested that robust optimization could be used in connection with traditional approaches to handle more complex cases. Other possibilities are to be considered. LDR is a crude approximation of the anticipation mechanism in the decision process. But is is already a highly complex one in regard of the actual decision process in the environmental and energy planning problems (and in many other problems!). Moreover, in an implementation phase of the model solution, the optimal LDR will almost surely never be implemented as such. Rather, at each time stage, a fresh version of the problem will be considered to account for the new situation. This leads to view the LDR not as a practical “open loop control”, but like a tool yielding a plausible anticipation of future evolution and providing valuable information in the design of the first stage decisions. 4

For instance, Markovian decision processes with reduced state space cardinality can be treated very efficiently by dynamic programming

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An interesting alternative, though probably a computationally costly one, would be to work in a folding horizon framework and scenario simulations. Suppose a scenario is selected, which prescribes up to the horizon the realization of the various uncertainties. The decision-maker only knows the probabilistic way the scenario is built, but not the scenario itself. The planning model may have a rather rich description of uncertainty in its first stage, in a robust optimization framework, and a cruder description for the later stages based on a LDR. The first stage optimal solution of this model is implemented in connection with the realization of the stochastic process in the first stage. This determines the state of the system at the beginning of the second stage, where a problem of the same nature as before, and a shorter horizon, is considered anew. This approach has been implemented in [2, 3, 6] and could prove interesting in more general models. The secondary goal of this chapter was to give a simple enough introduction to the burgeoning field of robust optimization. We deliberately confined our presentation to the linear programming context, except for a few mention of ellipsoidal uncertainty sets. In the latter case, the robust counterparts fall in the realm of second order cone programming (SCOP) for which highly efficient solution methods exist. The striking fact is that in the case of ellipsoidal uncertainty sets the robust counterpart is obtained via similar duality-based arguments. Further generalizations can be made that take advantage of the power of conic programming with linear constraints and self-dual cones. Most recent contributions in robust optimization are concerned with these extensions and/or use them intensively. They are beyond the scope of this introduction, but they cannot be ignored. The hope is that this introduction will make it easier for the reader to get around with the growing literature. Finally, we would like to mention that the probabilistic results in Section 8 have been considerably strengthened in some recent contributions. The crude hypotheses of Theorem 3 of random factors with zero mean and a symmetric range around the mean can be replaced by a range on the mean and a bound of the variance. Many other results of comparable nature are reported in [5]. They give more evidences of the strong link between chance constrained programming and robust optimization. Acknowledgment. The work was partially supported by the European research project Planets.

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